Geometry of Two-Variable Associated Legendre Polynomials

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On the left is a three-dimensional plot of a Legendre polynomial in two variables and ; on the right is a two-dimensional plot of the surface cut by a plane perpendicular to the axis. The exponential operator transform is defined in the Details.

Contributed by: Marcello Artioli and Giuseppe Dattoli (February 2016)
Open content licensed under CC BY-NC-SA



As the two-variable Legendre polynomials [1] can defined by the generating function [2]

. (1)

They reduce (Snapshot 1) to the two-variable Legendre polynomials [3] after the substitution

The Laplace transform identity


casts the RHS of equation 1 into the form


by setting and in equation 2. Recalling [4] that the generalized two-variable Hermite–Kampé de Fériet (H-KdF) polynomials


are generated through

we find that


and thus from equations 3 and 5 we get

. (6)

From the following property of the H-KdF polynomials,

equation 6 becomes

. (7)

Applying similar simplification used in [1], equation 4 also implies that

. (8)

Comparing 8 with 1, we find finally

By following the same criterion as in [4], we show the two-variable associated Legendre polynomials in a three-dimensional plot, displaying the relevant geometrical structure, and we have specified the polynomials determined by the intersection with a plane moving along the axis and parallel to the - plane.

Also, the operators defined in [1] can be generalized in the same way; we end up with



Accordingly, the operator transforms an ordinary monomial (when is 0) into a associated Legendre polynomial (when is not 0), and the plots represent the relevant geometrical interpretation. The monomial-polynomial evolution is shown by moving the cutting plane orthogonal to the axis: for a specific value of the polynomial degree, the polynomials lie on the cutting plane, as shown in the Snapshots.


[1] M. Artioli and G. Dattoli. "Geometry of Two-Variable Legendre Polynomials" from the Wolfram Demonstrations ProjectA Wolfram Web Resource.

[2] D. Babusci, G. Dattoli, and M. Del Franco, Lectures on Mathematical Methods for Physics, Rome: ENEA, 2011. 58_ENEA.pdf.

[3] L. C. Andrews, Special Functions for Engineers and Applied Mathematicians, New York: MacMillan, 1985.

[4] M. Artioli and G. Dattoli. "Geometric Properties of Generalized Hermite Polynomials" from the Wolfram Demonstrations ProjectA Wolfram Web Resource.

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